Question: Simplify; express your answer in exponential form. Assume $r\neq 0, n\neq 0$. $\dfrac{{(r^{3}n^{4})^{5}}}{{(rn^{5})^{-2}}}$
Answer: To start, try simplifying the numerator and the denominator independently. In the numerator, we can use the distributive property of exponents. ${(r^{3}n^{4})^{5} = (r^{3})^{5}(n^{4})^{5}}$ On the left, we have ${r^{3}}$ to the exponent ${5}$ . Now ${3 \times 5 = 15}$ , so ${(r^{3})^{5} = r^{15}}$ Apply the ideas above to simplify the equation. $\dfrac{{(r^{3}n^{4})^{5}}}{{(rn^{5})^{-2}}} = \dfrac{{r^{15}n^{20}}}{{r^{-2}n^{-10}}}$ Break up the equation by variable and simplify. $\dfrac{{r^{15}n^{20}}}{{r^{-2}n^{-10}}} = \dfrac{{r^{15}}}{{r^{-2}}} \cdot \dfrac{{n^{20}}}{{n^{-10}}} = r^{{15} - {(-2)}} \cdot n^{{20} - {(-10)}} = r^{17}n^{30}$